Nonparametric Functional Smoothing Techniques

一套人口测量数据的相对频率分布模型，称为概率密度函数，描述了落在不同间隔内的观测数据的比例，并回答了诸如"加拿大人中有多少百分比低于贫困线？或者"在装配线上生产的产品中有多少百分比低于规定的质量阈值？“这是一条数学曲线，比如钟形高斯曲线，然后用来计算人口的各种其他特征，比如平均值、中位数和平均值。通常，标准模型（称为参数模型，仅依赖于少数常数，例如高斯模型中的平均值和标准差）在计算总体测量的复杂函数时很有用。例如，在钟形曲线上，我们估计95%的观测值将落在平均值的两个标准差之内。如果模型不合适，这种基于模型的推论将无效。* * 在这种情况下，可以采用对基础分布进行宽松（或更少）假设的非参数模型。非参数方法背后的基本思想是对一组离散点的连续近似，通常称为平滑。因此，近似直方图的连续曲线可以用作非参数密度估计器。这个建议是在非参数平滑的一般领域，以期探索在各种应用领域可能具有重要意义的重要应用。* * 我已经开发和研究了非参数平滑方法，作为传统核平滑的替代方法，当对称核可能不合适时（例如当处理非标准数据时，包括加权数据，删失数据和相关数据）。这些方法依赖于非对称内核和离散分布，如二项分布和泊松分布，这些分布可用于估计样本中最大观测值以外的生存概率以及人口的其他重要特征，如给定年龄条件下的预期寿命。本提案的基本目标是探讨如何使用适用于上述非标准数据情况的方法来解决其他重要问题，如确定极端值和异常值、取决于基本概率密度函数的聚类和分类。* * 这些发展的意义在于它们的应用，例如在使用更新函数的估计量的保修分析中，以及在分类和聚类中，参数密度的作用被半/非参数对应物取代。该建议将进一步用于培训本科生和研究生在几个应用领域的非参数曲线平滑的知识和应用。

A model for the relative frequency distribution of a set of measurements on a population, known as a probability density function, describes the proportion of observations falling in various intervals, and answers such questions as "what percentage of Canadians are below the poverty line?" or "what percentage of items produced on an assembly line will be found below the prescribed quality threshold?" This is a mathematical curve, such as the bell-shaped Gaussian curve, that is then used to compute various other characteristics of the population such the average, median and percentiles. Often the standard models, known as the parametric models that depend only on a few constants, such as the mean and standard deviation in the case of the Gaussian model, are useful in computing complex functions of the population measurements. For example, on a bell curve, we estimate that 95% of the observations will fall within two standard deviations from the mean. If the model is not appropriate, such model based inferences will be invalid. ******In such situations, non-parametric model with relaxed (or less) assumptions about the underlying distribution may be employed. The basic idea behind the non-parametric method is a continuous approximation to a set of discrete points, commonly known as smoothing. Thus a continuous curve approximating the histogram may serve as a non-parametric density estimator. This proposal is in the general area of non-parametric smoothing, with a view to explore important applications that may be of importance in various applied fields. ******I have developed and studied non-parametric smoothing methods as an alternative to traditional kernel smoothing when symmetric kernels may not be appropriate (such as when dealing with non-standard data including weighted data, censored data and dependent data). These methods depend on asymmetric kernels and discrete distributions such as the binomial and Poisson distributions that are useful for estimating the survival probability beyond the largest observation in the sample as well as other important characteristics of the population such as the expected life time conditional on a given age. The basic objective of this proposal is to explore the use of methods that are applicable to non-standard data situations as mentioned above for other important problems such as the identification of extremes and outliers, clustering and classification which depend on the underlying probability density function. ******Significance of these developments is in their applications, for example in warranty analysis using the estimator of a renewal function, and in classification and clustering where the role of parametric densities is replaced by their semi/non-parametric counterparts. This proposal will be further used for training of undergraduate and graduate students in the knowledge and applications of non-parametric curve smoothing in several applied fields.